Hisham Altai. Definitions:

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1 University of Misan College of Engineering CONTROL Electrical Engineering Department Hisham Altai Definitions: 1- Control Systems: A system is a combination of components that act together to perform a specific goal. 2- Reference input: It is the actual signal input to the control system. 3- Controlled variable (output): the quantity that must be maintained at prescribed value. 4- Disturbance: An unwanted input signal that affects the output signal. Open-Loop control system: A system in which the output has no effect on the input action. In other words, the output is neither measured nor fed back for comparison with the input. One practical example is a washing machine. Disturbance Input Reference signal Summing point Plant Controlled variable (Output) Figure 1 Open-Loop Control System. 1

2 1- Closed-Loop Control System: A system in which the output has an effect on the input quantity in a way that can maintain the desired output value. An example is a room temperature control system. Forward Path Reference Error Signal Controller Contro l Signal Plant Controlled variable (Output) Measured Sensor Feedback Path Figure 2 Closed-loop control system Controller 2 Controller 1 Desired output response Inner loop Actual output Outer loop Sensor 1 Sensor 2 Figure 3 Multiloop feedback system with an inner loop and an outer loop. Closed-Loop versus Open-Loop Control Systems. Closed-Loop: The use of feedback makes the system response insensitive to external disturbances and internal variations in system parameters. More complicated and more expensive comparing with Open-Loop. Open-Loop The open-loop control system is easier to build because system stability is not a major problem. 2

3 It is sensitive to external disturbances. 2- Plants: The device, process, or system that need to be controlled. 3- Control unit (dynamic element): the unit that reacts to an actuating signal to produce a desired output. This unit does the work of controlling the output and thus may be a power amplifier. 4- Feedback control system: The unit that provides the means for feeding back the output quantity, or a function of the output, in order to compare it with the reference input. 5- Actuating signal: The signal that is difference between the reference input and the feedback signal if actuates the control unit in order to maintain the output of the desired value. 6- The sensor or measuring element is a device that converts the output variable into another suitable variable, such as a displacement, pressure, voltage, etc. 7- The actuator is a power device that produces the input to the plant according to the control signal so that the output signal will approach the reference input signal. 8- Automatic Controllers. An automatic controller compares the actual value of the plant output with the reference input (desired value), determines the deviation, and produces a control signal that will reduce the deviation to zero or to a small value. 3

4 Figure 4 Block diagram of an industrial control system, which consists of an automatic controller, an actuator, a plant, and a sensor The controller detects the actuating error signal, which is usually at a very low power level, and amplifies it to a sufficiently high level Examples of control systems Room temperature control system The output signal from a temperature sensing device such as a thermocouple or a resistance thermometer is compared with the desired temperature. Any difference or error causes the controller to send a control signal to the gas solenoid valve which produces a linear movement of the valve stem, thus adjusting the flow of gas to the burner of the gas fire. The desired temperature is usually obtained from manual adjustment of a potentiometer. 4

5 Figure 5 Room temperature control system Figure 6 Block diagram of room temperature control system. Steady conditions will exist when the actual and desired temperatures are the same, and the heat input exactly balances the heat loss through the walls of the building. The system can operate in two modes: a) Proportional control: Here the linear movement of the valve stem is proportional to the error. This provides a continuous modulation of the heat input to the room producing very precise temperature control. This is used for applications where temperature control, of say better than l C, is required (i.e. hospital operating theatres, industrial standards rooms, etc.) where accuracy is more important than cost. 5

6 b) On-off control: Also called thermostatic or bang-bang control, the gas valve is either fully open or fully closed, i.e. the heater is either on or off. This form of control produces an oscillation of about 2 or 3 C, of the actual temperature about the desired temperature, but is cheap to implement and is used for low-cost applications (i.e. domestic heating systems). Laplace transform In order to compute the time response of a dynamic system, it is necessary to solve the differential equations (system mathematical model) for given inputs. Laplace transform is one of the favored ways by control engineers to do this. This technique transforms the problem from the time (or t) domain to the Laplace (or s) domain. The advantage in doing this is that complex time domain differential equations become relatively simple s domain algebraic equations. When a suitable solution is arrived at, it is inverse transformed back to the time domain. L[f(t)] = f(t)e st dt = F(s) 0 Where s is complex variable σ ± jω and is called the Laplace operator. Example: Find Laplace transform for 1- f(t) = 1 2- f(t) = e at 1-6

7 L[f(t)] = 1e st dt 0 = [ 1 s (e st )] 0 = [ 1 s (0 1)] = 1 s 2- L[f(t)] = F(s) = e at e st dt 0 e (s+a)t dt 0 [ 1 s + a (e (s+a)t )]0 [ 1 (0 1)] s + a 1 s + a Derivatives: the Laplace transform of a time derivative is d n dt n f(t) = sn F(S) f(0)s n 1 f (0)s n 2 Where f(0), f (0) are the initial conditions, or the values of f(t), d f(t) etc. at t = 0. dt Example: equation given: Find the Laplace transform of the following differential (a) Initial conditions x o = 4, (b) Zero initial conditions dx 0 dt = 3 7

8 d 2 x o dt dx o dt + 2x o = 5 Solution (a) Initial conditions x o = 4, dx 0 dt = 3 (s 2 X o (s) 4s 3) + 3(sX o (s) 4) + 2X o (s) = 5 s (b) Zero initial conditions s 2 X o (s) + 3sX o (s) + 2X o (s) = 5 + 4s s (s 2 + 3s + 2)X o (s) = 5 + 4s2 + 15s s X o (s) = 4s2 + 15s + 5 s(s 2 + 3s + 2) At t = 0, x o = 0 dx o dt = 0 s 2 X o (s) + 3sX o (s) + 2X o (s) = 5 s X o (s) = 5 s(s 2 + 3s + 2) Table 1 Common Laplace transform pairs Time function f ( t ) Laplace transform L[f(t)] = F(s) 1- Unit impulse 1 2- Unit step 1 3- Unit ramp t 4- t n 1 s 1 s 2 n! s n+1 8

9 5- e at 6-1 e at 7- sin ωt 8- cos ωt 9- e at sin ωt 1 s+a 1 s+a ω s 2 +ω 2 s s 2 +ω 2 ω (s+ω) 2 +ω e at (cos ωt a ω sin ωt) s (s+ω) 2 +ω 2 Inverse transform f(t) = L 1 [F(s)] = 1 2πj σ+jω σ jω F(s)est ds In practice, inverse transformation is most easily achieved by using partial fractions to break down solutions into standard components, and then use tables of Laplace transform pairs 9

10 Table 2 Common partial fraction pairs Example: Find Laplace inverse transform for: 3s+2 s(s+1) Solution 3s + 2 s(s + 1) = A s + B s + 1 3s + 2 = A(s + 1) + Bs 3s + 2 = (A + B)s + A A = 2 10

11 L 1 [ 2 s + (2 + B) = 3 B = 1 1 ] = 2 + e t s + 1 Transfer function The function of a linear time invariant differential equation system is defined as the ratio of Laplace transform of the output( response function) to the Laplace transform of the input(drive function) under the assumption that all initial conditions are zero. Where: x is the input y is the output Transfer function = G(s) = L[output] L[input] zero intial conditions = Y(s) X(s) = b 0s m + b 1 s m b m 1 s + b m a 0 s n + a 1 s n b n 1 s + b n Block Diagrams. A block diagram of a system is a pictorial representation of the functions performed by each component and of the flow of signals. 11

12 Figure 7 Element of a block diagram. Summing Point: A circle with a cross is the symbol that indicates a summing operation. The plus or minus sign at each arrowhead indicates whether that signal is to be added or subtracted. Figure 8 Summing point Branch Point: A branch point is a point from which the signal from a block goes concurrently to other blocks or summing points. Open-Loop Transfer Function and Feedforward Transfer Function The ratio of the feedback signal B(s) to the actuating error signal E(s) is called the open-loop transfer function. That is, Open loop transfer function = B(s) E(s) = G(s)H(s) The ratio of the output C(s) to the actuating error signal E(s) is called the feedforward transfer function, so that Feedforward transfer function = C(s) E(s) = G(s) 12

13 If the feedback transfer function H(s) is unity, then the open-loop transfer function and the feedforward transfer function are the same. Closed-Loop Transfer Function. For the system shown above, the output C(s) and input R(s) are related as follows: since C(s) = G(s)E(s) E(s) = R(s) B(s) = R(s) H(s)C(s) Eliminating E(s) from these equations gives C(s) = G(s) [R(s) H(s)C(s)] C(s) R(s) = G(s) 1 + H(s)C(s) 13

14 Closed-Loop System Subjected to a Disturbance. Figure 9 Closed-loop system subjected to disturbance In examining the effect of the disturbance D(s), we may assume that the reference input is zero; we may then calculate the response C D (s) to the disturbance only. This response can be found from C D (s) D(s) = G 2 (s) 1 + G 1 (s)g 2 (s)h(s) On the other hand, in considering the response to the reference input R(s), we may assume that the disturbance is zero. Then the response C R (s) to the reference input R(s) can be obtained from C R (s) R(s) = G 1 (s)g 2 (s) 1 + G 1 (s)g 2 (s)h(s) C(s) = C D (s) + C R (s) References = G 2 (s) 1 + G 1 (s)g 2 (s)h(s) [G 1R(s) + D(s)] 1- Ogata, Katsuhiko, and Yanjuan Yang. "Modern control engineering." (1970): Burns, Roland. Advanced control engineering. Butterworth-Heinemann, Dorf, Richard C. Modern control systems. Addison-Wesley Longman Publishing Co., Inc.,